Template:ZPI/doc

This template generates a table for a particular EDO's zeta peak index.

Parameters

All parameters are required.

{{{steps}}}

The zeta-optimised number of steps per octave, corresponding to the location of a particular EDO's nearest local maximum or minimum according to the Riemann–Siegel Z function, [math]\displaystyle{ Z\left(\frac{2\pi}{\ln 2}x\right) }[/math]. For example, 12edo's local zeta-optimised step count is 12.0231830072926.

{{{step size}}}

The step size in cents, which is 1200 divided by the aformentioned zeta-optimised step count.

{{{tempered height}}}

The absolute value of Z evaluated at the zeta-optimised step count. For 12edo, this value is 5.193290 when evaluated at 12.0231830072926.

{{{pure height}}}

The absolute value of Z evaluated at the exact integer. For 12edo, this value is 5.084467 when evaluated at exactly 12.

{{{integral}}}

The absolute value of the integral between the two roots (zeroes) of Z closest to the given EDO. For 12edo, this value is 1.269599.

{{{gap}}}

{{{edo}}}

octave

The zeta-optimised octave size. For 12edo, this is 1197.68616940005.

consistent

The ZPI's consistency limit.

distinct

The ZPI's distinct consistency limit.

Example (12edo)

Here is the complete zeta peak index table for 12edo:

{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
			Tempered	Pure
34zpi	12.023183	99.807181	5.19329	5.084467	1.269599	15.899282	1197.686169	−2.313831	10	6